Pythagorean theorem

Pythagoras was born on the Aegean island of Samos Pythagoras was born on the Aegean island of Samos. He is best known for the Pythagorean Theorem, which relates the side lengths of a right triangle. a2 + b2 = c2

Legs = a and b (adjacent to the right angle) Hypotenuse = c (opposite the right angle)

Find the length of the hypotenuse to the nearest hundredth. c 4 5 a2 + b2 = c2 Pythagorean Theorem 42 + 52 = c2 Substitute for a and b. 16 + 25 = c2 Simplify powers. 41 = c2 41 = c Solve for c; c = c2. 6.40  c

Find the length of the hypotenuse to the nearest hundredth. triangle with coordinates (1, –2), (1, 7), and (13, –2) a2 + b2 = c2 Pythagorean Theorem 92 + 122 = c2 Substitute for a and b. 81 + 144 = c2 Simplify powers. 225 = c Solve for c; c = c2. 15 = c

Find the length of the hypotenuse to the nearest hundredth. 5 7 c a2 + b2 = c2 Pythagorean Theorem 52 + 72 = c2 Substitute for a and b. 25 + 49 = c2 Simplify powers. 74 = c Solve for c; c = c2. 8.60  c

Find the length of the hypotenuse to the nearest hundredth. Check It Out: Example 1B Find the length of the hypotenuse to the nearest hundredth. triangle with coordinates (–2, –2), (–2, 4), and (3, –2) x y (–2, 4) The points form a right triangle. a2 + b2 = c2 Pythagorean Theorem 62 + 52 = c2 Substitute for a and b. 36 + 25 = c2 Simplify powers. (–2, –2) (3, –2) 61 = c Solve for c; c = c2. 7.81  c

Solve for the unknown side in the right triangle to the nearest tenth. a2 + b2 = c2 Pythagorean Theorem 25 Substitute for a and c. b 72 + b2 = 252 49 + b2 = 625 Simplify powers. –49 –49 Subtract 49 from each side. b2 = 576 7 b = 24 Find the positive square root

Solve for the unknown side in the right triangle to the nearest tenth. a2 + b2 = c2 Pythagorean Theorem 12 b 42 + b2 = 122 Substitute for a and c. 16 + b2 = 144 Simplify powers. –16 –16 Subtract 16 from both sides. 4 b2 = 128 b  11.31 Find the positive square root.

Two airplanes leave the same airport at the same time Two airplanes leave the same airport at the same time. The first plane flies to a landing strip 350 miles south, while the other plane flies to an airport 725 miles west. How far apart are the two planes after they land? a2 + b2 = c2 Pythagorean Theorem 3502 + 7252 = c2 Substitute for a and b. 122,500 + 525,625 = c2 Simplify powers. 648,125 = c2 Add. 805 ≈ c Find the positive square root. The two planes are approximately 805 miles apart.

Two birds leave the same spot at the same time Two birds leave the same spot at the same time. The first bird flies to his nest 11 miles south, while the other bird flies to his nest 7 miles west. How far apart are the two birds after they reach their nests? a2 + b2 = c2 Pythagorean Theorem 112 + 72 = c2 Substitute for a and b. 121 + 49 = c2 Simplify powers. 170 = c2 Add. 13 ≈ c Find the positive square root. The two birds are approximately 13 miles apart.

Quiz Use the figure for Problems 1 and 2. 1. Find the height h of the triangle. 8 m 2. Find the length of side c to the nearest meter. 10 m c h 12 m 3. An escalator in a shopping mall is 40 ft long and 32 ft tall. What distance does the escalator carry shoppers? 6 m 9 m 2624 ≈ 51 ft